Problem: The number of degrees in the measures of the interior angles of a convex pentagon are five consecutive integers. What is the number of degrees in the largest angle?
Explanation: Suppose we list the angle measures in increasing order.  Let $x$ be the measure of the middle angle in degrees. Then, the five angles have measures $x -2^\circ$, $x-1^\circ$, $x$, $x + 1^\circ$, and $x+2^\circ$.  The sum of the angle measures in a pentagon is $180(5-2) = 540$ degrees, so \[(x -2^\circ)+(x-1^\circ)+(x)+(x + 1^\circ) +(x+2^\circ) = 540^\circ.\] Simplifying the left side gives $5x = 540^\circ$, and dividing both sides by 5 gives $x = 108^\circ$.  Therefore, the largest angle has measure $x + 2^\circ = \boxed{110^\circ}$.